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DietmarBauer

Departmentof ElectricalEngineering

Linkoping University, SE-58183 Linkoping,Sweden

WWW: http://www.control.isy.l iu.s e

Email: Dietmar.Bauer@tuwien.ac.at

June 6,2000

REG

LERTEKNIK

AUTO

MATIC CONTR

OL

LINKÖPING

Report no.: LiTH-ISY-R-2265

Submitted toCDC'2000, Sydney, Dec. 2000

TechnicalreportsfromtheAutomaticControlgroupinLinkopingareavailablebyanonymousftp

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DietmarBauer 

Institutef. Econometrics, OperationsResearch and System Theory,

TUWien, Argentinierstr. 8,A-1040 Wien

June 6,2000

Abstract

Inmoderndata analysisoften the rststepis to performsomedata preprocessing, e.g.

detrendingor eliminationofperiodiccomponentsofknownperiodlength. Thisis normally

done using leastsquares regression. Onlyafterwards black box models are estimatedusing

eitherpseudo-maximum-likelihoodmethods,predictionerrormethodsorsubspacealgorithms.

Inthispaperitisshown,thatfor subspacemethodsthisisessentiallythesameasincluding

the corresponding input variables, e.g. a constant or a trend or aperiodic component, as

additionalinputvariables. Hereessentiallymeans,thattheestimatesonlydi erthroughthe

choiceofinitialvalues.

Keywords: subspacealgorithms,linearsystems,estimation,identi cation

1 Introduction

Ithasbecomestandardinmoderntimeseriesanalysistoperformsomeform ofpreprocessingon

thedatapriortoidenti cation. Howeveroftenthee ectsofthispreprocessingarenotdealt with

in the identi cation phase. As an exampleconsider the analysis of so called subspacemethods:

Manyalgorithmshavebeenproposedbydi erentauthors,whichareallsubsumedunderthename

'subspacealgorithms',asthey showcertain similarities. Toname just themost popular wecite

CCA(Larimore, 1983), MOESP(Verhaegen, 1994)and N4SID(VanOverscheeand DeMoor,1994).

Thepropertiesof thesealgorithms havebeen analysedin anumberof papers(forreferencessee

e.g.Baueretal.,1999). Howeverallcitedreferencesrefertothecase,thatnopreprocessingprior

toidenti cationtakesplace. Alsonormallysomepersistencyconditionsontheinputareimposed,

which excludese.g. the constant as an inputvariable. Therefore essentiallyit is assumed, e.g.

that theonly sourcefor anonzero mean of the output is due to the ltered input. Whenthis

doesnotholdandnopreprocessingisperformed,abiaswillresult. Ifpreprocessingisperformed

the properties of thedata change, which has to be takeninto account forthe estimation of the

uncertainties, i.e. for the estimation of the asymptotic variances of the model obtained by the

identi cation step. This in many cases will be straightforward, however it seems to be more

convenient to perform both thepreprocessing and theactual identi cationin one step,in order

tounify thetreatmentandtosimplifythecalculationoftheasymptoticvariance.

Theaimofthispaperistoshow,thatthepreprocessingandtheestimationphaseinfactcanbe

uni edbyincludingtheconstantorsimilartermsasinputvariablesandusegeneralisedinverses

i.e. regularisationtechniquesintherespectiveregressions. Infact,itwillbeshown,thatthisleads

toessentiallythesameresultsinthesense,thatthereisnodi erenceintheattainedaccuracy,i.e.

that theasymptoticdistributionof theestimated systemareidentical. Thisshows,that someof

thepreprocessingmightbeincludedintothesubspacealgorithmwithoutanyproblem,and thus

canalsobeanalysedalongthesamelines.



Thiswork has been done, while the author was holdinga post doc positionat the Departmentfor

Auto-maticControl,UniversityofLinkoping,Sweden. The nancialsupportbythe EUTMRproject'SI'isgratefully

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arestated. Section3thengivesashortdescriptionofthemainstepsof theprocedure. Section4

then states the main results of the paper. In section 5 a numerical example is presented and

section6concludes.

2 Model set and assumptions

Inthispaperwedealwithdiscretetime, nitedimensional,timeinvariant,statespacesystemsof

theform x t+1 = Ax t +Bz t +K" t y t = Cx t +Dz t +" t (1) Herey t

denotesthesdimensionalobservedoutput,u

t

denotesthemdimensionalobservedinput,

"

t

the s dimensional white noise. A 2 R nn ;B 2 R nm ;C 2 R s n ;D 2 R s m ;K 2 R ns are

parameter matrices. The system usually will be described as (A;B;C ;D;K). We will assume

throughout, that "

t

is i.i.d. with zero mean and variance matrix > 0, having nite fourth

moments. Throughout wewillassumethat thesystemisstable, i.e. j

max

(A)j<1holds,where



max

denotes an eigenvalueof maximummodulus. It will also be assumed, that the systemis

strictlyminimum-phase,i.e. thatj

max

(A KC)j<1.

Correspondingtotheinputwewillassume,that theinput canbepartitionedinto twoparts:

u t 2 R mi and p t 2 R m mi . Here u t

accounts for the identi cation input, whereas p

t

denotes

theadditionalinputsdue to thepreprocessing. These additionalinputswill berestrictedto the

followingchoices:

 aconstantterm,i.e. p

t =1;8t

 aperiodiccomponent,i.e. p

t;1

=sin(!t+)forknown!2( ;]and. Inthiscasealso

thelaggedvariablep

t;2 =p

t 1;1

hastobeincluded

 atimetrend,i.e. p

t =t;8t

These choices include the typicalpreprocessing likedetrendingand eliminating periodic

compo-nentsof known periodicity. Naturally,all these termscould be used in combination. Especially

the inclusion of the trend makes the inclusion of a constant necessary in order to account for

unknowninitial e ects. Thekeyfeature ofthese inputsis that theyare persistent oforder one.

Thisisthereasonforincludingthelaggedtermfortheperiodiccomponents. Itwillbeclearfrom

thetext, why thisisneeded. Notethat in asimilar fashionalso morecomplicated preprocessing

canbe dealt with, aslong asit is done using regressiononto deterministic processes which are

persistentoforderone.

Note, thatthe model asstatedis notidenti able forthese inputs, which canbeseenfrom a

discussionoftheconstant: Assumethattheinputhasnonzeromean

u

andtheoutputhasmean

 y . Thenweobtain  y =D u + 1 X j=0 CA j B u

Thusitis clear,that theconstributionoftheconstantisequalto D

p + P 1 j=0 CA j B p , whereB p andD p

denotethecolumnsofB andD respectivelycorrespondingtotheconstant. Thisleadsto

alinearequationinB

m andD

m

showingthenonidenti ability,whichwillcomplicatetheanalysis

oftheasymptoticdistribution.

Inthefollowingwewill always distinguishbetweenthese twokinds ofinputs mainly for the

reasonofstatingassumptionsfortheidenti cationinputs. Thereasonisthedi erentnatureofthe

inputs: All preprocessinginputsusedabovehavethecharacteristicofbeingperfectlypredictable

fromoneobservation,i.e. theyarepersistentlyexcitingofdegreeone. Fortheidenti cationinputs

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Subspaceprocedures havebeendescribedin anumberof papers. Thereforewerestrictourselves

to a short descriptiononly. Fordetails see the survey in (Bauer, 1998, Chapter 3). Let Y + t;f = [y 0 t ;;y 0 t+f 1 ] 0

denotethestackedvectorofoutputs,wheref isauserde nedinteger. Similarily

de ne U + t;f and E + t;f

using the inputs and the noise respectively in the place of the outputs.

Furthermore letZ t;p =[y 0 t 1 ;u 0 t 1 ;;y 0 t p ;u 0 t p ] 0

, where pis auserde ned integer. Then itis

easytoshowthefollowingequation:

Y + t;f =O f K p Z t;p +O f  A p x t p +U f U + t;f +E f E + t;f : (2) Here  A=(A KC),  B=(B KD),O f =[C 0 ;A 0 C 0 ;:::;(A f 1 ) 0 ] 0 and K p =[[K ;  B];  A [K ;  B];:::;  A p 1 [K ;  B] FurtherE f

denotestheblockToeplitzmatrix,whosei-thblockrowisequalto

[CA i 2 K ;:::;CK ;I s ;0 s(f i)s ] where I s

is the s-dimensional identity and 0 ab

is a ab dimensional nullmatrix. Finally U

f

denotestheblockToeplitzmatrix,whosei-thcolumn isequalto

[CA i 2

B;:::;CB;D;0

s(f i)m

]

This equation is the starting pointfor all subspacealgorithms. An outline canbe given as

follows: 1. RegressY + t;f ontoZ t;p andU + t;f

inorder toobtainestimates ^ z and ^ u . 2. ^ z

willtypicallybeoffullrank,whereasthelimit

z isofrankn. ThusaSVDof ^ W + f ^ z ^ W p = ^ U ^  ^ V 0 = ^ U n ^  n ^ V 0 n + ^

Risperformedleadingtoanestimate ^ O f =( ^ W + f ) 1 ^ U n ; ^ K p = ^  n ^ V 0 n ( ^ W p ) 1 . Here ^ W + f and ^ W p

are weightingmatrices, which haveto be chosen by theuser. Wewill

assume throughout that they are nonsingular (a.s.) and converge to some limit W + f and W + p respectively forT !1. ^  n

contains then dominantsingular valuesand ^ U n ; ^ V n the

correspondingleftorrightrespectivelysingularvectors.

3. Eithertheestimates ^ O f and ^ u ortheestimate ^ K p

isusedtoestimatethesystem.

In the last step, one hastwo choices: MOESPand one variant of N4SIDuse the structure in O

f

and

u

in order toestimate thesystemmatrices, whereasCCAandanothervariantof N4SIDuse

^

K

p

toestimatethestateasx^

t = ^ K p Z t;p

andthenusethesystemequations(1)in ordertoobtain

estimatesofthesystem.

Nowtherearetwopossiblewaystoinclude thepreprocessinginthisframework:

 Regressy t andu t ontop t

andlettheresidualsbedenotedasy~

t andu~

t

. This preprocessed

datathencanbeusedinthesubspacemethodin orderto obtainanestimateofthesystem

(A;B;C ;D;K).  Use y t and z t =[u 0 t ;p 0 t ] 0

asthe datafor thesubspacemethod to obtainan estimateof the

system(A;B;C ;D;K)ofthetransferfunction relatingu

t toy

t .

The aim of the next section is to show that these two procedures deliver essentially identical

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Note, thatin thepresentsetuptheregressionin the rststeptoobtaintheestimates ^ z ; ^ u will

havetotakeintoaccountthesingularityofthematrixoftheregressorsduetothemulticollinearity

introducedbytheprocessp

t . ForZ

t;p

note, thatin theactualcalculationonlythevectorsZ ;

t;p

areused fortheestimate ^ z . Here Z ; t;p

denotestheresidualsof Z

t;p regressedonto U + t;f . Using thenotationha t ;b t i=T 1 P T f t=p+1 a t b 0 t thisisequalto Z t;p hZ t;p ;U + t;f ihU + t;f ;U + t;f i y U + t;f Here y

denotes the Moore Penrose pseudo inverse. Due to the nature of the variables p

t , the

residuals in all coordinates corresponding to p

t

are zero. Note that this corresponds to using

the preprocessed data y~

t ;u~

t

up to the di erence of at most f +p terms in the calculation of

theregressionsforthepreprocessing. Thisdi erenceisasymptoticallynegligibleunder theusual

assumptionsonf andptobeo((logT) a

)forsomea<1. Thereforethecolumnsoftheestimate

^ z ,whichcorrepondtoy t oru t

respectively,areequivalenttotheonesobtainedfrompreprocessed

data. Theremainingcolumnscorrespondtop

t

andtheseareessentiallyzero(i.e. zerouptoe ects

duetoinitialvaluesintheregressions). Alsonotethatthecalculationofthepseudoinversecould

becircumventedby usingonlythose coordinatesof U +

t;f

,whichare linearilyindependent. These

can be found easily, since the dependence structure of p

t

is known. In fact, if the regression

is performed accordingto theformula given above,all coordinates corresponding to p

t+j ;j >0

simplycanbeomittedwithoutchangingtheresult.

IntheMOESPtypeofproceduresthisshowsthat theestimates ^

O

f

ofthetwoalternativeswill

beessentiallyidentical. Thisshowstheequivalenceoftheestimates ^

Aand ^

C. Fortheestimation

ofB andD weconsideraspecialalgorithmsimilarto theMOESPtypeofprocedures: LetU

f ( ^ O f )

denotetheestimateofU

f

,whereelementsofO

f

arereplacedbytheirrespectiveestimates. Note

thatU f ( ^ O f

)islinearinBandD. ThentheestimateofBandDcanbeobtainedfromminimising

k ^ O ? f  hY + t;f ;U + t;f ihU + t;f ;U + t;f i y U f ( ^ O f )  k Fr Here ^ O ? f ^ O f =0and ^ O ? f

isoffullrowrank. Fromthisitfollowsfromtheblockmatrixinversion,

that the columns correspondingto u

t

are identical to therespectivecolumns obtainedfrom the

preprocesseddata y~

t ;u~

t

. Fromthis itfollowsthattheestimation ofthepartofB andD, which

corresponds to u

t

is estimated essentiallyidentical asif the preprocesseddata would havebeen

used. This shows, that in this case theestimates of the coordinatesof thesystem (A;B;C ;D)

corresponding to u

t

is essentiallythesameasobtainedfrom thepreprocessed data. Concerning

thepartdue top

t

howeverthesameresultdoesnotseemtohold,comparetheexamplegivenin

section5. Thisisamatteroffurtherresearch.

FortheLarimoretypeofapproachwehaveseenthattheestimate ^

z

isessentiallyidenticalto

theestimateobtainedfromthepreprocesseddatainthecoordinatescorrespondingtotheoutputor

theinputduetou

t

,theothercomponentsbeingzero. Thereforethesameholdstruefor ^

K

p

,noting

that fortheusual weightingmatrices ^ W p ; ^ W p

alsoonlythedata Z ; t;p and Y +; t;f is used. Here Y +; t;f isde nedanalogouslyto Z ; t;f

. Thesevectorsareessentiallyidentical tothecorresponding

vectorsformed from the preprocesseddata. As aconsequencethe stateestimate x^

t = ^ K p Z t;p is

almostidenticaltotheestimateobtainedfromthepreprocesseddataexceptfordi erencesinthe

spacespannedbythecomponentsofp

t

. Notethatthenextstepinthisclassofalgorithmsistouse

thestate in aregression,wherethese deterministic componentsare included. This againshows,

thattheestimateofthesystemmatricescorrespondingtou

t

coincideuptoinitialconditionswith

theestimateobtainedfromthepreprocesseddata. Note,that thenonidenti ability inthis setup

iscircumvented,asthedeterministiccomponentsinx^

t+1 andx^

t

convergetosomevectors  x and x respectively. Letting u

denote thecoeÆcients ofthe deterministiccomponentsof z

t

then it

followsfromthestateequationthat

 x =A x +B u

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t

any nonidenti ability problems by construction and the estimates for B

p

, the columns due to

the deterministic components, are consistent and asymptotically normal. After the estimation

thenonidenti abilityofthemodelstructurecould beused toobtainadi erentnormalisationas

B

p

=0usingthestructureofthenonidenti ability. Inbothcasestheasymptoticdistributionwill

bethesame,asifthecorrespondingestimateshavebeencalculatedusing thepreprocesseddata,

asisstraightforwardtosee. Finallywehaveobtainedthefollowingtheorem:

Theorem1 Lety

t

be generatedbya systemofthe form(1), where the noise isi.i.d. with nite

momentsup tofourth order. Let the input z

t

consist of twodi erent components: z

t =[u 0 t ;p 0 t ] 0 , where u t

is aquasistationary sequence, which is persistently exciting of order f+p. Further p

t

iseither a constant term,a trend component ora periodic component with known frequency and

shift. Then theestimatesofG

i

(q),the transferfunctionrelatingu

t toy

t

obtainedbytheinclusion

of p

t

in the MOESP type of subspace methods and the estimates obtained by using the residuals

from a regression of y t andu t onto p t

as the data input for the subspace method without p

t are

asymptotically equivalent, i.e. the asymptoticdistributionof the estimationerrorare identical.

Iftheinputu

t

isanARMAprocessgeneratedbyi.i.d. whitenoisehavingauniformelybounded

spectrumandp=p(T)=o((logT) a );p(T) dlogT 2logj 0 j

,thenthesameholdsfortheLarimoretype

ofalgorithms. Inthiscasealsotheestimatesofthetransferfunctionrelatingthenoisetotheoutput

willbeasymptoticallyequivalent. Furthermorealsotheestimates ofthee ectsofthe deterministic

termsareessentially the same.

Itshould benoted,that theassumptionsof thetheorem arebynomeans necessary. Infact,

much weakerconditionssuÆce. Fora discussionon the necessaryconditionsin the MOESPtype

ofalgorithms see(Bauer andJansson, 2000),forassumptions inamartingaleframeworkfor the

Larimoretypeofproceduressee(Baueretal.,1999).

Thesigni canceof thetheorem is that itgivestheusertwodi erent but equivalentwaysto

deal with nonzero means, driftsand periodic componentsincluded in manytime series. Either

one canpreprocess the data and estimate and test for the inclusion of these terms beforehand

andthenusethepreprocesseddatainorder toobtainestimatesofthedynamicalsystems,which

arethemaingoalin manyapplications. Thealternativewayisto usethedeterministictermsas

additionalinputsin thesubspacemethod. Bothprocedureshaveadvantagesand disadvantages.

The big advantage for the preprocessing approach lies in the fact, that the testing procedures,

whetherthedetmerinisticcomponentsarecontainedinthedataathand,arewellestablishedand

implementedin many programs. This facilitatestheanalysis fortheuser. Ontheotherhand in

thiscasetheinclusionofthepreprocessingstagesinthecalculationoftheasymptoticvarianceof

thesubspaceestimatesispossible,although cumbersome.

Theinclusionofthetermsintheestimationprocedureiscomputationallysimple. Theoriginal

procedurescanbeused,iftheregressionsaredoneinarobustwayusingthepseudoinversesinthe

caseofmulticollinearities. Thedirectembeddingofthepreprocessingmakesthecalculationofthe

varianceeasier,althoughatthepresenttimenoprogramstocalculatetheasymptoticvarianceof

the subspaceprocedure exists, that could be used in an industrial context. Forthe MOESPtype

ofproceduresthederivation of theasymptoticdistribution oftheestimates ofthecolumns ofB

and D corresponding to p

t

seemsto becomplicated. HoweverfortheCCA typeofestimates the

derivation of these distributions is straightforward. Tests for more realistic hypotheses can be

obtainedfrom theasymptotic distribution,e.g. whether ornot thedeterministiccomponentsin

theoutputcanbeexplainedonlythroughthecomponentsoccuringintheinput.

5 Numerical Exmaple

Inthissectionwewillpresentsomesimpleexamples,whichillustratethetheorygivenin thelast

section. The rstexampleisaonestateSISOsystem,whichisdescribedbythefollowingmatrices:

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MOESP nopre. 0.9183 1.1498 0.0599 pre. 0.4983 0.9996 0.0013 incl. 0.4985 1.0001 0.0013 corr. 0.9983 0.9980 0.9978 CCA nopre. 0.8886 1.2125 0.0710 pre. 0.4999 0.9994 0.0011 incl. 0.5002 0.9998 0.0011 corr. 0.9970 0.9974 0.9974

Table1: ThistableshowsthemeanvaluesoftheestimatesofA;B andD estimatedwithsample

sizeT =1000usingMOESPandCCAontheoriginaldata(nopre.),onthepreprocesseddata(pre.)

orincludingaconstantterm(incl.) inthecalculations. Alsothecorrelation (corr.) betweenthe

estimates(pre.) and(incl.) isgiven.

TheinputisGaussianwhitenoisewithmean1andvariance1. Thenoise"

t

ischosento bezero

meanGaussianwhitenoiseofvariance1. Theoutputy

t isequalto y t =(D+C(q A) 1 B)u t +(1+C(q A) 1 K)" t +10

Here q denotes the forward shift operator. 1000 data sets of dimension T = 1000 have been

generatedandforeachdatasetthesystemisestimated

 using nopreprocessingandnoinclusionofaconstant,

 meancorrecteddataand

 using theoriginaldatawiththeadditionofaconstantinputvariable.

f =p=5hasbeenchosenfortheMOESPtypeofprocedureandf =5;p=15fortheLarimoretype

ofprocedure,inallthetrials. Aftertheestimationthesystemshavebeentransformedtoechelon

canonicalform. InbothcasestheCCAweightingshavebeenused. Themeanofthecorresponding

estimatesfortheMOESPtypeofprocedures canbeseeninTable1.

Itcanbeseenclearly,thattheestimateswithouttakingthenonzeroconstantsintoaccountis

biased. It alsocanbeseen,thattheothertwoapproachesgivecomparablemeansandarehighly

correlated,asisexpectedfromthelastsection.

Comparingthemean ofy

t

accordingto theestimated model tothe actuallyestimated mean

on the other hand shows a di erent picture: For the CCA procedure a correlation of 0:9989 is

obtained,i.e. almostperfectagreement,whilefortheMOESPprocedurethecorrelationisestimated

as0:0948showingalmost no lineardependence. This reaÆrmsthe complicationsto be awaited

forthederivation oftheasymptoticdistribution ofthe estimatesofthecomponentsofB and D

correspondingtop

t .

Asasecondexamplethesamesystemisused,buttheinputischangedbyaddingarandomly

weighteddeterminsticcomponentandalsotheoutputiscontaminatedbyasimilarterm:

u t = v t + x t x t = 2 6 6 6 6 6 6 4 t sin(0:5t) sin(0:22t) 1 sin(0:5(t 1)) sin(0:22(t 1)) 3 7 7 7 7 7 7 5

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0

20

40

60

80

100

120

140

160

180

200

−5

0

5

10

15

OUTPUT #1

0

20

40

60

80

100

120

140

160

180

200

−1

0

1

2

3

4

INPUT #1

Figure1: Observations1to200ofatypicaltrajecory.

Hereeachcomponentof isdrawnfroma[0;1]uniformdistribution. Also

y t =G(q)u t +H(q)" t + x t where G(q) = D+C(qI A) 1 B H(q) = I+C(qI A) 1 K

Thecomponentsof areuniformely[0;1]distributed. Finallyv

t and"

t

areGaussian zeromean

and unit variance random processes. 1000 time series with these characteristicsof sample size

T =1000areconstructed. Partsofatypicalinput-outputtrajectorypaircanbeseeninFigure1.

Theestimationshowsthesamepictureasthe rstexample:Figure2showsthestandarddeviations

ofthe estimatesofthe transferfunction D+C(qI A) 1

B at50 pointsin theanguar frequency

range[;]. Herebyf =p=5wasused andtheMOESPtypeof procedures. Theplotshowsthe

standarddeviationoftheMOESPestimatesusingthepreprocesseddataandthestandarddeviation

of the di erence between the estimates obtained from the preprocessed data and the estimates

obtained using the original data and including the additional variables, respectively. It canbe

seen clearly, that the di erence in between the twoapproaches is of lowermagnitude than the

estimation errors themselves. For the CCA case the results are almost identical, therefore the

presentationoftheresultsisomitted.

6 Conclusions

In this paper we did derive the asymptotic properties of subspace procedures, when the data,

whichisusedfortheprocedure,ispreprocessedbyremovingtrendsandperiodiccomponents. It

has been shown, that the preprocessingcan alternatively beseen asthe inclusion of additional

inputterms. Thismakesitpossibletocalculatetheasymptoticvarianceoftheestimatesobtained

withtheprocedurefromthestandardproceduresforthesubspaceestimates. Alsotestprocedures

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−1

0

1

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Ang. frequ. /

π

Standard Deviation

Comparison of Standard Deviations

MOESP

Diff.

Figure2: Theplotsshowthestandarddeviationofthetransferfunctionestimatesobtainedfrom

thepreprocesseddataandf =p=5(|) andthestandarddeviationofthedi erenceofthetwo

di erentprocedures(-+-).

References

Bauer,D.(1998).SomeAsymptoticTheoryfortheEstimationofLinearSystemsUsingMaximum

LikelihoodMethodsorSubspaceAlgorithms.PhDthesis.TUWien.

Bauer, D. and M.Jansson (2000).Analysis of theasymptoticpropertiesof theMOESP typeof

subspacealgorithms.Automatica36(4),497{509.

Bauer, D., M. Deistler and W. Scherrer (1999). Consistency and asymptotic normality of some

subspacealgorithmsforsystemswithoutobservedinputs.Automatica35,1243{1254.

Larimore,W. E. (1983).System identi cation, reduced order ltersand modeling viacanonical

variateanalysis.In: Proc.1983Amer.ControlConference2.(H.S.RaoandP.Dorato,Eds.).

Piscataway,NJ.pp.445{451. IEEEServiceCenter.

Van Overschee, P. and B. DeMoor(1994). N4sid: Subspace algorithms for the identi cation of

combineddeterministic-stochasticsystems.Automatica30, 75{93.

Verhaegen,M. (1994).Identi cation of thedeterministic partofmimo statespace modelsgiven

References

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